The average daily high temperature $($ in $^\circ C)$ in Karachi, Pakistan, on the $t^{\text{th}}$ day of the year is approximately $ T(t) = 29 - 5\cos\left(\dfrac{2\pi (t- 12)}{365}\right)$. What are the highest and lowest average daily highs in Karachi? Give exact answers. Highest:
Solution: Since $\cos x$ fluctuates between $-1$ and $1$, the maximum and minimum values of $\cos\left(\dfrac{2\pi (t-12)}{365}\right)$ are $1$ and $-1$, respectively: they occur when $\dfrac{2\pi (t-12)}{365} = 0$ or $\pi$. That means the extreme values of $- 5\cos\left(\dfrac{2\pi t}{365}\right)$ are $-5$ times as big, or $-5$ and $5$. (We switch which one is the maximum and which one is the minimum). The biggest $T(t)$ could be is when $- 5\cos\left(\dfrac{2\pi (t-12)}{365}\right) = 5$ and $\begin{aligned} T(t) &= 29- 5\cos\left(\dfrac{2\pi t}{365}\right) \\\\ &= 29+5 \\\\ &= 34 \end{aligned}$ The smallest $T(t)$ could be is when $- 5\cos\left(\dfrac{2\pi (t-12)}{365}\right) = -5$ and $\begin{aligned} T(t) &= 29- 5\cos\left(\dfrac{2\pi t}{365}\right) \\\\ &= 29-5 \\\\ &= 24 \end{aligned}$ The lowest high temperatures in Karachi are around $24^\circ C$, and they occur near $t = 12$, or the beginning of the year. The highest high temperatures in Karachi are around $34^\circ C$, half a year after $t = 12$.